公式大全
| Topic | Formula | Notes |
|---|---|---|
| Permutation | Order matters | |
| Combination | Order doesn't matter | |
| Permutations with repetition | Repetition allowed | |
| Combinations with repetition | Repetition allowed | |
| Binomial k-th term | ||
| Stars and Bars | Non-negative integer solutions | |
| Arithmetic explicit | — | |
| Arithmetic recursive | ||
| Geometric explicit | — | |
| Geometric recursive | ||
| Arithmetic series sum | — | |
| Geometric series sum (finite) | ||
| Geometric series sum (infinite) | ||
| Product Theorm | $$r_1 r_2 \left( \cos(x_1 + x_2) + i\sin(x_1 + x_2) \right)$$ | Multiplying two complex numbers in polar form |
| Quotient Theorem | $$\frac{r_1}{r_2} \left( \cos(x_1 - x_2) + i\sin(x_1 - x_2) \right)$$ | Dividing two complex numbers in polar form |
| De Moivre's Theorem | $$\left[ r \left( \cos x + i\sin x \right) \right]^n = r^n \left( \cos(nx) + i\sin(nx) \right)$$ | Raising a complex number in polar form to a power |
| nth Root Theorem | $$\sqrt[n]{r} \left( \cos\frac{x + 2k\pi}{n} + i\sin\frac{x + 2k\pi}{n} \right)$$ | For k = 0, 1, 2, ..., n-1, giving n distinct roots |
| Odd/Even Polynomial End Behavior & Root Multiplicity | Odd degree, leading coefficient > 0: bottom-left → top-right Odd degree, leading coefficient < 0: top-left → bottom-right Even degree, leading coefficient > 0: top-left → top-right Even degree, leading coefficient < 0: bottom-left → bottom-right |
Odd multiplicity root: cross the x-axis Even multiplicity root: bounce (touch but don't cross) |
| Order of Transformations | Stretch/Compress/Reflect first, then Shift | Order matters; different order yields different results |
| Sinusoidal Function Formula | $$y = a \sin(b(x - h)) + k \quad \text{or} \quad y = a \cos(b(x - h)) + k$$ | |a| = amplitude, period = 2π/|b|, h = horizontal shift, k = vertical shift |
| Exponential Function | $$y = a \cdot b^{x-h} + k$$ | b > 0, b ≠ 1; horizontal asymptote: y = k |
| Logarithmic Function | $$y = a \log_b (x-h) + k$$ | b > 0, b ≠ 1; vertical asymptote: x = h |
| Composite Argument Property | $$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$$$$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$$$$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$$ | Sum and difference formulas |
| Double Angle | $$\sin(2\theta) = 2\sin\theta \cos\theta$$ $$\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$$$$\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}$$ |
Derived from composite argument formulas with A = B = θ |
| Half Angle | $$\sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}}$$ $$\cos\frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}}$$$$\tan\frac{\theta}{2} = \frac{\sin\theta}{1 + \cos\theta} = \frac{1 - \cos\theta}{\sin\theta}$$ |
Sign depends on the quadrant of θ/2 |
| Pythagorean Identities | $$\sin^2\theta + \cos^2\theta = 1$$ $$\tan^2\theta + 1 = \sec^2\theta$$ $$1 + \cot^2\theta = \csc^2\theta$$ |
Derived from the unit circle and dividing by sin²θ or cos²θ |
| Cofunction Identities | $$\sin\left(\frac{\pi}{2} - \theta\right) = \cos\theta$$ $$\cos\left(\frac{\pi}{2} - \theta\right) = \sin\theta$$ $$\tan\left(\frac{\pi}{2} - \theta\right) = \cot\theta$$ $$\csc\left(\frac{\pi}{2} - \theta\right) = \sec\theta$$ $$\sec\left(\frac{\pi}{2} - \theta\right) = \csc\theta$$ $$\cot\left(\frac{\pi}{2} - \theta\right) = \tan\theta$$ |
Sine and cosine of complementary angles are equal |
| Odd/Even Identities | $$\sin(-\theta) = -\sin\theta$$ $$\cos(-\theta) = \cos\theta$$ $$\tan(-\theta) = -\tan\theta$$ $$\csc(-\theta) = -\csc\theta$$ $$\sec(-\theta) = \sec\theta$$ $$\cot(-\theta) = -\cot\theta$$ |
Sine, tangent, cosecant, cotangent are odd; cosine and secant are even |
| Law of Sines | $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ | Relates sides and opposite angles in any triangle |
| Law of Cosines | $$a^2 = b^2 + c^2 - 2bc\cos A$$ $$b^2 = a^2 + c^2 - 2ac\cos B$$ $$c^2 = a^2 + b^2 - 2ab\cos C$$ |
Generalization of Pythagorean theorem for any triangle |
| Polar to Rectangular Conversion | $$x = r\cos\theta$$ $$y = r\sin\theta$$ |
Converts polar coordinates (r, θ) to rectangular coordinates (x, y) |
| Rectangular to Polar Conversion | $$r = \sqrt{x^2 + y^2}$$ $$\theta = \tan^{-1}\left(\frac{y}{x}\right)$$ |
Converts rectangular coordinates (x, y) to polar coordinates (r, θ); adjust θ based on quadrant |
| Matrix Multiplication | Row of first matrix × Column of second matrix | Multiply corresponding entries, then sum; A(m×n) × B(n×p) = C(m×p) |
| Determinant of 2×2 Matrix | $$\det\begin{pmatrix} a & b \ c & d \end{pmatrix} = ad - bc$$ | Product of main diagonal minus product of other diagonal |
| Determinant of 3×3 Matrix (Quick Method) | $$\begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)$$ | Expand along first row; copy first two columns to the right, sum diagonals ↘ minus ↙ |
| Row Echelon Form (REF) | 1. All zero rows at the bottom 2. Leading entry (pivot) of each nonzero row is 1 3. Each pivot is to the right of the pivot above it |
Use Gaussian elimination; pivot must be 1, entries below pivot become 0 “从上到下，从左往右” |
| Reduced Row Echelon Form (RREF) | 1. All conditions of REF 2. Each pivot is the only nonzero entry in its column 3. Pivot columns form the identity matrix |
Use Gauss-Jordan elimination; entries above and below pivot all become 0 “从上到下，从左往右” |
| Inverse of 2×2 Matrix | $$A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}$$ | Requires ad - bc ≠ 0; swap a and d, negate b and c, divide by determinant |
| Parabola | Opens Up: $$(x - h)^2 = 4p(y - k), \quad p > 0$$ Opens Down: $$(x - h)^2 = -4p(y - k), \quad p > 0$$ Opens Right: $$(y - k)^2 = 4p(x - h), \quad p > 0$$ Opens Left: $$(y - k)^2 = -4p(x - h), \quad p > 0$$ |
Vertex: (h, k); Focus and Directrix depend on direction and p |
| Ellipse | Horizontal Major Axis: $$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1, \quad a > b$$ Vertical Major Axis: $$\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1, \quad a > b$$ |
Center: (h, k); c² = a² - b²; Foci and Vertices depend on major axis orientation |
| Hyperbola | Horizontal Transverse Axis: $$\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$$ Vertical Transverse Axis: $$\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$$ |
Center: (h, k); c² = a² + b²; Vertices, Foci, and Asymptotes depend on transverse axis orientation |
| Change of Base Formula | $$\log_a b = \frac{\log b}{\log a}$$ | Converts log of any base to common log (base 10) or natural log (base e) |
| Area of Triangle (SSS) — Heron's Formula | $$s = \frac{a + b + c}{2}$$ $$A = \sqrt{s(s - a)(s - b)(s - c)}$$ |
s = semi-perimeter; a, b, c are the three side lengths |
| Area of Triangle (SAS) | $$A = \frac{1}{2}ab\sin C$$ | a and b are two sides, C is the included angle between them |
| Direction Angle of a Vector | $$\theta = \tan^{-1}\left(\frac{y}{x}\right)$$ | Angle measured counterclockwise from the positive x-axis; adjust θ based on quadrant |
| Magnitude of a Vector | $$||\vec{v}|| = \sqrt{x^2 + y^2}$$ | For vector (x, y); also called length or norm |
| Unit Vector | $$\hat{u} = \frac{\vec{v}}{||\vec{v}||} = \left( \frac{x}{||\vec{v}||}, \frac{y}{||\vec{v}||} \right)$$ | Vector with magnitude 1 in the same direction as v⃗ |